Chapter 9
Trigonometric Equations
Equations in which the trig ratios of unknown angles are involved are called "Trigonometric Equations" such as the ones shown below:
sinx + cosx = 0.75 and tan^{2}x  3tanx = 0.
The solution (s) to a trig equation are angles that satisfy the equation. For example, one of the solutions to the trig equation: cos2x + sinx = 1 is x = π/6. To verify, let's substitute π/6 for x in the equation as follows:
cos (π/3) + sin(π/6) = 1 or,
1/2 + 1/2 =1 or,
1 = 1.
To solve the trig equations, we keep simplifying them until they reduce to one of the following forms:
sinx = sinβ, or cosx = cosβ, or, tanx = tanβ, and we use the method of Chapter 8, to write down the possible solution sets.
Example 1:
Solve the trig
equation: sinx + tanx cos(2π  x) = 1.

Example 2: Solve the trig equation given below and find all solutions between 0 and 2π.
2sin^{2}x + 3cosx = 3.
We got two possible sets of answers for each cosine equation; however, in set (1):
x = k(2π) ± 0 is just x = 2kπ.
In set (2): we may write it as x = 2kπ ± π/3.
The problem is asking for answers between 0 and 2π.
Setting k = 0, (1) yields: x = 0, and (2) yields: x = +π/3. Note that x = π/3 is out of range.
Setting k = 1, (1) yields: x = 2π, and (2) yields: x = 2ππ/3. x = 2π +π/3 is out of range.
Finally, the collection of the desired answers are: x = 0, π/3, 5π/3, and 2π.
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Example 3: Solve the trig equation given below and find all solutions between 0 and 2π.
tanx + cotx = 2.
For k = 0, the solution is x = π/4.
For k = 1, the solution is x = 5π/4.
For any other k value, the solution will be out of the desired range.
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Exercises: Solve the following trigonometric equations: